We derived block Extended **Trapezoidal** **Rule** of First (ETRs) that are A-stable of up to order 8. The methods were shown tocompete favorablythan other existing methodsin terms of accuracy (see tables 1 and 2). Our newly derived methods in block form are shown to have extensive regions of stability and in particular are A-stable up to order 8 and so very suitable for stiff system of ordinary differential equations. The continuous formulation for each step number k is evaluated at the end point of the interval to recover the individual main method for FETR, SETR and EETR in (13) and their counterparts schemes in (8), (11) and (12) respectively, to this end, the idea of additional conditions is discarded. Furthermore, we do not need any pair in our implementation. Our block methods preserve the A-stability property of the **trapezoidal** **rule** (refer to figure 1), they are also less expensive in terms of the number of functions evaluation per step.

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As can be seen in Figure 1 and Figure 2, the S transformation converts the spherical Bessel inte- grals into a simpler sine integral, but the integrand has slow convergence (Figure 2). The convergence is greatly improved by applying a DE variable transformation x = Mφ(τ) (Figure 3). Because the transformation induces rapid convergence to zero, sinc quadrature (**trapezoidal** **rule**) renders highly e ffi cient approximation, and very few collocation points are required to obtain accurate results when using either transformation φ 1 (τ) or φ 2 (τ) (17). Usually, less collocation points are needed when using

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This paper attempts to propose a method of assessing learner’s competencies using fuzzy **trapezoidal** **rule**. While calculating learner’s competencies, learner’s other professional performances are being considered besides skill attributes. Developed prototype of the ITS shows that the learners can successfully track their competencies without the help of others because of the use of the automated messaging about their development.

We develop self-starting family of three and five step continuous extended trapezoidal rule of second kind a block hybrid type (BHETR 2 s) methods through interpolation an[r]

We combine suitable arithmetic average approximations, with explicit backward Euler formula, and derive a third-order L-stable derivative-free error-corrected **trapezoidal** **rule** LSDFECT . Then, we apply LSDFECT **rule** to the linearized Burgers’ equation with inconsistent initial and boundary conditions and test its stability and exactness. We use Mathematica 7.0 for computation.

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The following corollary which provides an upper bound for the approximation error in the trapezoid quadrature formula, for f of bounded variation, holds [4].. Corollary 2.[r]

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N = 198, 200, 250 and this leads to values of ∆t ≃ 0.0051, 0.0050, 0.0040 and the Courant (CFL) numbers ν = ∆x ∆t = 200 N ≃ 1.0101, 1, 0.8. It can be seen easily from the Figure 1 and Figure 3 that midpoint method and **trapezoidal** **rule** perform well up to CFL numbers =1 but their results quickly deteriorate when applied with larger CFL numbers. Therefore, in practical sense these two methods are equally eﬃcient with regard to TVD. With other RK2 methods ( 1 2 ≤ κ ≤ 1), a similar behavior was observed (see Figure 2). The necessity of the step size restriction (2.4) was experimentally studied for several RK2 methods with κ > 1, (see Figure 4). In general it was found that the (2.4) is somewhat too strict.

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Abstract. Inequalities are obtained for weighted integrals in terms of bounds involving the first derivative of the function. Quadrature rules are obtained and the classical Iyengar inequality for the **trapezoidal** **rule** is recaptured as a special case when the weight function w (x) ≡ 1. Applications to numerical integration are demonstrated.

The structure of the paper is the following. Section 2 contains some auxiliary results of independent interest, needed in the sequel. In Lemma 1, we prove an interesting inequal- ity, which may be seen as a reﬁned version of a reversed Cauchy–Bunyakovsky–Schwarz inequality (see Remark 2). In Lemma 3, we give bounds for the error of approximation of an integral by **trapezoidal** **rule** in terms of the ﬁrst derivative for the function (which complements the asymptotic error estimates, valid just for large values of the parameter), a useful practical result in numerical analysis. Lemma 4 is a technical result concerning the sign of a certain function, essential for proving our main results.

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To investigate these issues further, Figure 3 provides some plots showing how estimates of standard normal SRMs vary with different quadrature methods and different values of N. These plots are based on an illustrative ARA coefficient equal to 5, but we get similar plots for other values of this coefficient. The methods examined are those based on the **trapezoidal** **rule**, Simpson’s **rule**, and Niederreiter and Weyl quasi-Monte Carlo. As we might expect, all four quadrature methods give estimates that converge on their true values as N gets larger. However, the **trapezoidal** and Simpson’s **rule** methods produce estimates that converge smoothly as N gets larger, whereas the two quasi-Monte Carlo methods produce estimates that converge more erratically as N gets larger. The plots also suggest that the first two methods are usually more accurate for any given value of N, and that the method based on Simpon’s **rule** is marginally better than that based on the **trapezoidal** **rule**.

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The area under the absorbance vs wavelength curve for experimental samples between 340nm and 400nm was calculated using trapezoidal rule according to equation 6: Area = [ Abs340+ Abs370 [r]

Abstract. It is well-known that the **trapezoidal** **rule**, while being only second-order accurate in general, improves to spectral accuracy if applied to the integration of a smooth periodic function over an entire period on a uniform grid. More precisely, for the function that has a square integrable derivative of order r the convergence rate is o N −(r−1/2)

The main aim of this paper is to point out various estimates for I p ( f ) using an approach which employs the removal of the singularity and the utilisation of the trapezoidal rule.. So[r]

Richardson’s Extrapolation: These methods use two estimates to get more accurate approximation. The estimate and error associated with a multiple- application **trapezoidal** **rule** can be represented as, ---------(1) where I is the exact value of the integral. I(h) is approximation from an n-segment application of **trapezoidal** **rule** and h = (b-a)/n is step size.

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Perturbed **trapezoidal** type rules are obtained in Section 5.3 using what are termed as premature variants of Gr¨ uss, Chebychev and Lupa¸s inequalities. Atkin- son [30] uses an asymptotic error estimate technique to obtain what he defines as a corrected **trapezoidal** **rule**. His approach, however, does not readily produce a bound on the error.

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Integral inequalities have been used extensively in most subjects involving mathematical anal- ysis. They are particularly useful for approximation theory and numerical analysis in which estimates of approximation errors are involved. In this paper, by the use of an integral iden- tity, we point out some new integral inequalities for the **trapezoidal** **rule** and apply these to special means: p-logarithmic means, logarithmic means, identric means etc., and in numerical integration.

where x x 0 , 1 and is usually distinct from the appearing in equation (7). This is the **trapezoidal** **rule** plus error term. The integral is approximated by 1 2 h f 0 f 1 . The **Trapezoidal** **rule** is usually applied in a composite form. To estimate the integral of f over (a, b), we divide (a, b) into N sub-intervals of equal length h b a N . The end points of the sub-intervals are x 1 a ih i , 0,1,..... N , so that x 0 a and x N b .

now provide a proof when K(t) is piecewise smooth with (finite) jump discontinuities irrespective of where the jumps occur. In particular, convergence does not rely on fitting or adapting the stepsize so that the jumps occur at element boundaries, in contrast to the requirements of the **trapezoidal** **rule** (collocation with continuous piecewise linear approximation of u) applied to (1.1) with a step function kernel [5, subsection 4.2.2] and methods for second kind problems in, e.g., [3, Chapter 4.2] and [13].

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Figure 10. Step local error for **trapezoidal** **rule**, compared to RK4 and explicit Euler methods. Notice that, despite the large step size in the **trapezoidal** **rule** method, it does not currently run faster than RK4. The reasons are the necessity to solve a system of nonlinear algebraic equations at each step using an iterative Newton-like method, and the need to compute the Jacobian matrix. We omit details here, but in short, the number of Newton’s iterations is low (2–3) when the Jacobian matrix is recomputed at each iteration, and is significantly higher (10–20) in cases with approximated Jacobian matrix (e.g., using an update formula similar to the Broyden’s one [7]).

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From this ﬁeld diagram, calculate the area of the grassed region using the trapezoidal rule.. (All measurements are in metres.).[r]

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